Showing papers on "Elementary function published in 1998"
TL;DR: Using an alternate representation of the Marcum Q-function, an expression for the bivariate Rayleigh cumulative distribution function is found in the form of a single integral with finite limits and an integrand composed of elementary functions.
Abstract: Using an alternate representation of the Marcum Q-function, an expression for the bivariate Rayleigh cumulative distribution function is found in the form of a single integral with finite limits and an integrand composed of elementary functions. This result has advantage over previous forms of the same CDF which involve the Marcum Q-function itself or are expressed as infinite series of products of integrals.
55 citations
TL;DR: A new and unified methodology for computing first order derivatives of functions obtained in complex multistep processes is developed on the basis of general expressions for differentiating a composite function, and the formulas for fap automatic differentiation of elementary functions are derived.
Abstract: A new and unified methodology for computing first order derivatives of functions obtained in complex multistep processes is developed on the basis of general expressions for differentiating a composite function. From these results, we derive the formulas for fap automatic differentiation of elementary functions, for gradients arising in optimal control problems, nonlinear programming and gradients arising in discretizations of processes governed by partial differential equations. In the proposed approach we start with a chosqn discretization scheme for the state equation and derive the exact gradient expression. Thus a unique discretization scheme is automatically generated for the adjoint equation For optimal control problems, the proposed computational formulas correspond to the integration of the adjoint system of equations that appears in Pontryagin's maximum principle. This technique appears to be very efficient, universal, and applicable to a wide variety of distributed controlled dynamic systems an...
49 citations
02 Oct 1998
TL;DR: In this paper, the authors propose a method that allows fast evaluation of these functions in double precision arithmetic, using small tables, small multipliers, and for some functions, a final 'large' multiplication.
Abstract: This paper deals with the computation of reciprocals, square roots, inverse square roots, and some elementary functions using small tables, small multipliers, and for some functions, a final 'large' multiplication. We propose a method that allows fast evaluation of these functions in double precision arithmetic.The strength of this method is that the same scheme allows the computation of all these functions.
48 citations
TL;DR: This article describes a collection of Fortran routines for multiple-precision complex arithmetic and elementary functions that provide good exception handling, flexible input and output, trace features, and results that are almost always correctly rounded.
Abstract: This article describes a collection of Fortran routines for multiple-precision complex arithmetic and elementary functions. The package provides good exception handling, flexible input and output, trace features, and results that are almost always correctly rounded. For best efficiency on different machines, the user can change the arithmetic type used to represent the multiple-precision numbers.
28 citations
TL;DR: In this article, the known model of nonlinear dispersive waves, which was proposed by Boussinesq in the second half of the nineteenth century, was considered and solutions of the Bousseinq equation, which are expressed via elementary functions and describe wave packets, their interaction between each other and with solitons, and some other structures were obtained.
Abstract: The known model of nonlinear dispersive waves, which was proposed by Boussinesq in the second half of the nineteenth century, is considered. Solutions of the Boussinesq equation, which are expressed via elementary functions and describe wave packets, their interaction between each other and with solitons, and some other structures are obtained. To construct these solutions, Hirota's bilinear representation and differential relations specified by ordinary differential equations with constant coefficients are used.
27 citations
TL;DR: In this article, the convergence characteristics of these two formulae are discussed and compared with numerical quadratures of M and S. The results are amenable for use in further theoretical studies of groundwater percolation and mounding.
Abstract: M. S. Hantush established relationships between the dynamics of groundwater mounding beneath recharge zones and two integral functions, M and S. Exact algebraic expressions for these functions are developed in terms of a formal power series expansion. This expansion may be reordered to provide two independent analytical partial summations involving elementary functions. The convergence characteristics of these two formulae are discussed and compared with numerical quadratures of M and, hence, S. The algebraic expressions are used to generate identities for related integrals. Compact algebraic approximations to M and S can be deduced from the series expansions with essentially arbitrary accuracy, while retaining valuable functional information. For example, a simple two-term truncated sum yields a reasonable approximation to M over a useful range of arguments. The results are amenable for use in further theoretical studies of groundwater percolation and mounding where numerical quadratures may be undesirable.
16 citations
TL;DR: In this article, the spherical-ellipsoidal Stokes function is constructed in a closed form for the boundary value problem with ellipsoid corrections in the boundary condition for anomalous gravity.
Abstract: Green's function for the boundary-value problem of Stokes's type with ellipsoidal corrections in the boundary condition for anomalous gravity is constructed in a closed form. The `spherical-ellipsoidal' Stokes function describing the effect of two ellipsoidal correcting terms occurring in the boundary condition for anomalous gravity is expressed in O(e
2
0)-approximation as a finite sum of elementary functions analytically representing the behaviour of the integration kernel at the singular point ψ=0. We show that the `spherical-ellipsoidal' Stokes function has only a logarithmic singularity in the vicinity of its singular point. The constructed Green function enables us to avoid applying an iterative approach to solve Stokes's boundary-value problem with ellipsoidal correction terms involved in the boundary condition for anomalous gravity. A new Green-function approach is more convenient from the numerical point of view since the solution of the boundary-value problem is determined in one step by computing a Stokes-type integral. The question of the convergence of an iterative scheme recommended so far to solve this boundary-value problem is thus irrelevant.
10 citations
TL;DR: In this article, the Landau-Lifshitzitz equation for a spin chain with an easy plane in the case of spin non-flip is solved by the method of inverse scattering transform.
Abstract: The Landau-Lifshitz equation for a spin chain with an easy plane in the case of spin non-flip is solved by the method of inverse scattering transform. To avoid complexity caused by the Riemann surface of the usual spectral parameter, a particular parameter k is introduced. After performing a gauge transformation corresponding to , the resulting Lax pair is independent of particular solutions in this limit. An inverse scattering transform is then developed in terms of k. A system of linear equations is derived in the reflectionless case. An expression of the gauge transformation and hence expressions of multi-soliton solutions are found explicitly by using the Binet-Cauchy formula. As an example, an explicit expression of the 1-soliton is given in terms of elementary functions of x and t.
5 citations
Posted Content•
TL;DR: An analytical-numeric calculation method of extremely complicated integrals is presented in this paper, where the appropriate analytical continuation and a corresponding integration contour allow to reduce the calculation of wide class of integrals to a numeric search of integrand denominator roots (in a complex plane) and subsequent residue calculations.
Abstract: An analytical-numeric calculation method of extremely complicated integrals is presented. These integrals appear often in magnet soliton theory. The appropriate analytical continuation and a corresponding integration contour allow to reduce the calculation of wide class of integrals to a numeric search of integrand denominator roots (in a complex plane) and a subsequent residue calculations. The acceleration of series convergence of residue sum allows to reach the high relative accuracy limited only by roundoff error in case when $10÷15$ terms are taken into account. The circumscribed algorithm is realized in the C program and tested on the example allowing analytical solution. The program was also used to calculate some typical integrals that can not be expressed through elementary functions. In this case the control of calculation accuracy was made by means of one-dimensional numerical integration procedure.
2 citations
Journal Article•
TL;DR: In this paper, the authors derived the field and energy density in a plate where the magnetization vector is an arbitrary function of all coordinates and periodic in two directions in the plane of the plate.
Abstract: For the first time, equations are derived using a nontraditional fast method for the field and energy density in a plate where the magnetization vector is an arbitrary function of all coordinates and periodic in two directions in the plane of the plate. The results in special cases agree with those reported to date [1-4]. The energy density is derived for tilted domains. For a periodic stripe structure, the solution is radically simplified, since it is represented in terms of the integrals of magnetization and elementary functions (without an expansion into a series). This leads to a number of new applications and simplifies the calculation of stripe structures.
2 citations
TL;DR: In this paper, the nonrelativistic retarded Schrodinger Green's function was derived for electron propagation in a rotating coordinate system subject to a constant, uniform axial magnetic field (B = B k ), including the effect of a Pauli spin term in the Hamiltonian.
02 Oct 1998
TL;DR: A new Composite Polynomial Method for generating elementary functions using second-order interpolation, which takes the average of two second-degree polynomials with an overlapping subinterval, and features the minimum approximation error the overlapping sub interinterval.
Abstract: High-speed elementary function generation is crucial to the performance of many DSP applications. This paper presents a new Composite Polynomial Method for generating elementary functions using second-order interpolation. The composite polynomial takes the average of two second-degree polynomials with an overlapping subinterval, and features the minimum approximation error the overlapping subinterval. Its implementation uses small memory with minimal additional computational circuitry and represents a competitive design with respect to existing designs.